In this paper it is shown that if Ω⊂RN\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega \subset \mathbb {R}^N$$\end{document} is an open, bounded Lipschitz set, and if f:Ω×Rd×N×N→[0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$f: \varOmega \times \mathbb {R}^{d \times N \times N} \rightarrow [0, \infty )$$\end{document} is a continuous function with f(x,·)\documentclass[12pt]{minimal}
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\begin{document}$$f(x, \cdot )$$\end{document} of linear growth for all x∈Ω\documentclass[12pt]{minimal}
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\begin{document}$$x \in \varOmega $$\end{document}, then the relaxed functional in the space of functions of Bounded Hessian of the energy F[u]=∫Ωf(x,∇2u(x))dx\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F[u] = \int _{\varOmega } f(x, \nabla ^2u(x)) dx \end{aligned}$$\end{document}for bounded sequences in W2,1\documentclass[12pt]{minimal}
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\begin{document}$$W^{2,1}$$\end{document} is given by F[u]=∫ΩQ2f(x,∇2u)dx+∫Ω(Q2f)∞(x,dDs(∇u)d|Ds(∇u)|)d|Ds(∇u)|.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\mathcal {F}}}[u] = \int _\varOmega {{\mathcal {Q}}}_2f(x, \nabla ^2u) dx + \int _\varOmega ({{\mathcal {Q}}}_2f)^{\infty }\bigg (x, \frac{d D_s(\nabla u)}{d |D_s(\nabla u)|} \bigg ) d |D_s(\nabla u) |. \end{aligned}$$\end{document}This result is obtained using blow-up techniques and establishes a second order version of the BV relaxation theorems of Ambrosio and Dal Maso (J Funct Anal 109:76–97, 1992) and Fonseca and Müller (Arch Ration Mech Anal 123:1–49, 1993). The use of the blow-up method is intended to facilitate future study of integrands which include lower order terms and applications in the field of second order structured deformations.
